Two open organ pipes of fundamental frequencies `n_(1) and n_(2)` are joined in series. The fundamental frequency of the new pipes so obtained will be

Options

A. `(n_(1)+n_(2))/(n_(1)n_(2))`
B. `(n_(1)n_(2))/(2n_(2)+n_(1))`
C. `(2n_(2)+n_(1))/(n_(1)n_(2))`
D. `(n_(1)n_(2))/(n_(1)+n_(2))`

Answer

D

Explanation

For open pipe with frequency `n_(1)` and length `I_(1)`,
`n_(1)=v/(2I_(1))rArrI_(1)v/(2n_(1))` ……(i)
Similarly, for open pipe with frequency `n_(2)` and length `I_(2)`,
`n_(2)=v/(2I_(2))rArrI_(2)=v/(2n_(2))` …(ii)
When both the pipes are joined together then net length of the pipe will be,
`L=I_(1)+I_(2)` ...(iii)
Let, after joining the pipes fundamental frequency will be `N_(0)`,
So, `L=v/(2N_(0))=I_(1)+I_(2)rArrL=v/(2n_(1))+v/(2n_(2))=v/(2n_(0))`
Using Eqs. (i) and (ii), we get
`rArr N_(0)=(n_(1)n_(2))/(n_(1)+n_(2))`